The Heine-Borel Theorem states that a subset of Euclidean space is compact if and only if it is both closed and bounded. It gives the standard working definition of compactness in finite-dimensional spaces and is named for Eduard Heine and Emile Borel, who established special cases of the result before it was stated in full generality.
Facts
StatementA subset of Euclidean space is compact if and only if it is both closed and bounded. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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In Branch
Proved By
Source Heine-Borel Theorem (Wikipedia)
Sources
1. Heine-Borel Theorem (Wikipedia)
Wikimedia FoundationHistory
Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine-Borel theorem.
Lead section
In real analysis in mathematics, the Heine-Borel theorem, named after Eduard Heine and Émile Borel, states:
Proved By: Peter Gustav Lejeune Dirichlet, Lead
Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.
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